psychologywikiaorg-20200213-history
Cram
In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F^* compared to a given empirical distribution function F_n , or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as : \omega^2 = \int_{-\infty}^{\infty} F_n(x)-F^*(x)^2\,\mathrm{d}F^*(x) In one-sample applications F^* is the theoretical distribution and F_n is the empirically observed distribution. Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case. The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928-1930. The generalization to two samples is due to Anderson.Anderson (1962) The Cramér–von Mises test is an alternative to the Kolmogorov-Smirnov test. Cramér–von Mises test (one sample) Let x_1,x_2,\cdots,x_n be the observed values, in increasing order. Then the statistic is Pearson & Hartley (1972) p 118 : T = n \omega^2 = \frac{1}{12n} + \sum_{i=1}^n \left[ \frac{2i-1}{2n}-F(x_i) \right]^2. If this value is larger than the tabulated value the hypothesis that the data come from the distribution F can be rejected. Watson test A modified version of the Cramér–von Mises test is the Watson testWatson (1961) which uses the statistic U''2, where : U^2= T-n( \bar{F}-\tfrac{1}{2} )^2, where : \bar{F}=\frac{1}{n} \sum F(x_i). Cramér–von Mises test (two samples) Let x_1,x_2,\cdots,x_N and y_1,y_2,\cdots,y_M be the observed values in the first and second sample respectively, in increasing order. Let r_1,r_2,\cdots,r_N be the ranks of the x's in the combined sample, and let s_1,s_2,\cdots,s_M be the ranks of the y's in the combined sample. Anderson shows that : T = N \omega^2 = \frac{U}{N M (N+M)}-\frac{4 M N - 1}{6(M+N)} where U is defined as : U = N \sum_{i=1}^N (r_i-i)^2 + M \sum_{j=1}^M (s_j-j)^2 If the value of T is larger than the tabulated values, the hypothesis that the two samples come from the same distribution can be rejected. (Some books give critical values for U, which is more convenient, as it avoids the need to compute T via the expression above. The conclusion will be the same). The above assumes there are no duplicates in the x , y , and r sequences. So x_i is unique, and its rank is i in the sorted list x_1,...x_N . If there are duplicates, and x_i through x_j are a run of identical values in the sorted list, then one common approach is the ''midrank Ruymgaart (1980) method: assign each duplicate a "rank" of (i+j)/2 . In the above equations, in the expressions (r_i-i)^2 and (s_j-j)^2 , duplicates can modify all four variables r_i , i , s_j , and j . See also *Nonparametric statistical tests Notes References * * *Pearson, E.S., Hartley, H.O. (1972) Biometrika Tables for Statisticians, Volume 2, CUP. ISBN 0521069378 (page 118 and Table 54) *Ruymgaart, F. H., (1980) "A unified approach to the asymptotic distribution theory of certain midrank statistics". In: Statistique non Parametrique Asymptotique, 1±18, J. P. Raoult (Ed.), Lecture Notes on Mathematics, No. 821, Springer, Berlin. *Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle", Biometrika, 48 (1/2), 109-114 Further reading * External links * C-vM Two Sample Test (Documentation for performing the test using R * Table of Critical values for 1 sample CvM test Category:Statistical tests Category:Statistical distance measures Category:Nonparametric statistical tests Category:Normality tests